Geometrical interpretation of the Casimir effect
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Geometrical interpretation of the Casimir effect
Eugene B. Kolomeisky and Joseph P. Straley, Department of Physics, University of Virginia, P. O. Box 400714, Charlottesville, Virginia 22904-4714, USA Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA
Casimir forces are a manifestation of the change in the zero-point energy of the vacuum causedby the insertion of boundaries. We show how the Casimir force can be efficiently computed byconsideration of the vacuum fluctuations that are suppressed by the boundaries, and rederive thescalar Casimir effects for a series of the Dirichlet geometries. For the planar case a finite universalforce is automatically found. Consistent with other calculations of the effect, for curved geometriesdivergent (non-universal) expressions are encountered. They are interpreted geometrically followingCandelas and Deutsch (1979) as largely due to the divergent self-energy of the boundary contributingto the force. This viewpoint is supported by explicit calculations for a wedge-circular arc geometry intwo dimensions where non-universal and universal contributions into the effect can be unambiguouslyseparated. We also give a heuristic derivation of the purely geometrical expression (Sen, 1981) forthe non-universal piece of the Casimir energy due to an arbitrary smooth two-dimensional Dirichletboundary of a compact region.
PACS numbers: 03.70.+k, 11.10.-z, 11.10.Gh, 42.50.Pq
I. INTRODUCTION
Casimir interactions are due to the macroscopic re-sponse of the physical vacuum to the introduction ofboundaries. They were first derived as an attractiveforce between perfectly conductive parallel plates inducedby the zero-point motion of the electromagnetic field [1]There is convincing experimental evidence for the realityof these forces [2] and a vast body of literature dedicatedto various aspects of the phenomenon [3].The Casimir interaction E is the difference between thevacuum energy of the system constrained by the bound-aries and that of free space. Since boundaries made ofreal materials are transparent to sufficiently high-energymodes [1], the high energy spectrum is unaffected bythe geometry of the system, and only a finite range ofthe spectrum need be considered. However, in the theo-retical treatments of this effect the vacuum energies areusually calculated from an effective low-energy harmonicfield theory (such as quantum electrodynamics in the caseof the electromagnetic Casimir effect), so that they areapproximated by the sum of zero-point energies of a col-lection of simple harmonic oscillators with a spectrum ω = c | k | (where c is the speed of light). In this model,the dispersion relation holds for arbitrarily large wavevectors k ; both the ”constrained” and ”free” vacuum en-ergy densities are ultraviolet divergent; and the Casimirinteraction is the difference between two infinite quanti-ties. The theory resolves this problem by a soft-cutoffmodification of the large- k part of the spectrum thatleads to a finite vacuum energy. The result is not verysensitive to the form of the cutoff, so that the Casimirinteraction can then be extracted by taking the cutoffto infinity at the end of calculation. Other approachesto the calculation that make use of analytic continuation[4] and dimensional regularization [5] techniques give thesame answer, thus adding to the credibility of the result.As already noted, the divergences are more mathemat- ical artifacts than physical reality. The important virtueof the model is that for many geometries a finite univer-sal result depending only on ~ , c , and macroscopic lengthscales is obtained without introducing a cutoff. However,this is not always the case: specifically, the divergencesoccurring for spherical geometry in even space dimen-sions do not cancel [6, 7]. What this means physicallyrepresents an open problem; it seems to imply that intwo dimensions a conducting ring placed in vacuum isunstable.The goal of this paper is two-fold: first, we show howthe Casimir effect can be efficiently computed by directconsideration of the fluctuation modes that are elimi-nated by the presence of the boundaries. Technicallythis is accomplished by using a method closely related tothe path-integral approach to the Casimir effect [8].Second, we compute the Casimir effect in a two-dimensional wedge-circular arc geometry. The resultsof our analysis lend support to the idea of Candelasand Deutsch [9] that the divergences encountered in thecase of curved boundaries are of geometrical nature andmostly due to divergent self-energy of the boundary con-tributing to the Casimir force. Since the geometry westudy is a relative of the ring geometry in two dimen-sions, our results also shed some light on the physicalmeaning of the divergent Casimir force exerted on thering in two dimensions [6, 7].The organization of this paper is as follows. In SectionI we introduce a method of computation of the Casimireffect which is based on consideration of the vacuum fluc-tuations eliminated by introduction of boundaries. Theefficiency of this technique is illustrated in Section IIwhere the scalar Casimir energies and pressures are re-derived for a series of standard geometries: planar ge-ometry in an arbitrary dimension, circular geometry intwo dimensions, cylindrical and spherical geometries inthree dimensions, and finally, spherical geometry in anarbitrary space dimension. We observe that the Casimirpressures are always divergent in curved geometries. Fol-lowing Candelas and Deutsch [9] we pursue an interpre-tation of these divergences as non-universal contributionsdue to the geometry of the boundary. This is made ex-plicit in detailed calculations of Section III where theeffect is computed for the wedge-circular arc geometryin two spatial dimensions with the wedge edges subjectto the Dirichlet and periodic boundary conditions. Thelatter includes the circular geometry in two dimensionsas a special case. Our main observation here is that non-universal contributions to the Casimir effect can indeedbe attributed to the geometry of the boundary. Moreoverthe non-universal pieces of the effect turn out to be insen-sitive to the change of the topology of the vacuum whilethe universal ones are. This fact is used in Section IV toheuristically re-derive a purely geometrical formula [10]for the non-universal piece of the Casimir energy of an ar-bitrary smooth Dirichlet boundary of a compact region.We conclude (Section V) by summarizing our findings. II. FORMALISM
In what follows we will be analyzing Casimir effects ina Gaussian field theory with the Euclidian action S [ w ] = 12 Z ~ /T dτ d d x (cid:18) c − ( ∂w∂τ ) + ( ∇ w ) (cid:19) , (1)where T is the temperature. The real scalar w is a func-tion of the d -dimensional position vector r and imagi-nary time τ , and is periodic on the Matsubara circle, w ( r ,
0) = w ( r , ~ /T ) [11]. The action (1) is applicable atenergies low compared to some scale ~ ω ; the frequencycutoff ω is provided by the properties of the materialthe boundary is made of.The scalar field theory (1) can be viewed as a toy ver-sion of electricity and magnetism. The divergences en-countered for curved geometries exist in both theories, sothey are not due to specifics of electricity and magnetism.Therefore we study the problem in the simpler setting ofthe scalar theory.The zero-point energy can be calculated by means ofa functional integral that makes use of the correspon-dence between the Feynman path integral for the d -dimensional field theory and the partition function fora d + 1-dimensional classical statistical mechanics prob-lem. The path integral is Z w = Z Dw ( r , τ ) exp( − S [ w ] / ~ ) (2)but it can be interpreted as the partition function for aclassical statistical mechanics problem with the Hamil-tonian S at a fictitious temperature which is equal toPlanck’s constant [12]. The zero-point energy corre-sponds to the ”free energy” per unit ”length” in the imag-inary time direction, so that E = − ~ (ln Z w ) / ( ~ /T ) = − T ln Z w . Now assume that the vacuum is disturbed by the ad-dition of sharp boundaries which constrain the field insome way and thus eliminate some degrees of freedom ofthe vacuum fluctuations. The constrained field (which wewill refer to as v ) inherits any boundary conditions im-posed on w as well as new conditions on surfaces D i and N j of Dirichlet ( v | D i = 0) or Neumann ( ∂v/∂n | N j = 0)type, respectively, where the subscripts i and j label theboundaries and ∂v/∂n is the normal derivative. We willwrite the difference between the original and constrainedfields in the form w ( r , t ) = v ( r , t ) + u ( r , t ), where u ( r , t )can be chosen to satisfy the d + 1-dimensional equation( ∂ c ∂τ + △ ) u = 0 , u | D i = f i ( r , τ ) , ∂u∂n | N j = g j ( r , τ ) (3)where f i and g j are functions defined on the boundariesand determined by the boundary values of w ; they playthe role of dynamical variables of our approach. Thereason for defining u this way is that it eliminates thecross term in the action, so that S [ w ] = S [ v ]+ S [ u ]. Thenthe functional integral factors into integrations over v and u , so that Z w = Z v Z u . Since the zero-point energy for theunconstrained system is determined by Z w and the zero-point energy for the constrained system is determinedby Z v , the Casimir energy is given by E = T ln Z u . Thisresult allows us to calculate directly the change in energydue to the fluctuation modes that have been eliminated.A simplification is achieved by expanding all the dy-namical variables of the problem into a Fourier seriesin the imaginary time domain; for example u ( r , τ ) = P ω u ω ( r ) exp iωτ where the Fourier coefficients u ω ( r )are solutions to the boundary-value problem for theHelmholtz equation( △− ω c ) u ω = 0 , u ω | D i = f ω,i ( r ) , ∂u ω ∂n | N j = g ω,j ( r ) (4)The calculation of the action S [ u ] is further simplifiedwhen the identity ( ∇ u ) = div ( u ∇ u ) − u △ u is substi-tuted into Eq.(1). Then the integral of div ( u ∇ u ) over d d x transforms into a sum of surface integrals. The re-maining integral over dτ vanishes due to the relation △ u = − ∂ u/c ∂τ and the condition of periodicity, u ( r ,
0) = u ( r , ~ /T ). As a result we find S [ u ] = 12 Z ~ /T dτ X i Z [ u ∇ u ] i d s i = ~ T X ω,i Z [ u ω ∇ u − ω ] i d s i (5)Here [ ψ ] i stands for the discontinuity of ψ across the i -thboundary, and the summation is performed over all theboundaries.Although our approach is applicable to an arbitrarynumber of the Dirichlet and/or Neumann boundaries, inall the cases considered in this paper only one Dirichletboundary contributes into the integral (5). Since we aredealing with a harmonic field theory, the solution to theboundary-value problem (4) is linear in the surface field f ω and thus the action S E is a quadratic diagonal formof f ω : S E = ~ T X ω,ν | f ων | λ ν ( | ω | /c ) (6)where the subscript ν (representing one or more indices)labels the normal modes of the field u that have beeneliminated by the boundary in question, and λ ν ( | ω | /c ) > E = T X ω,ν ln 2 πT λ ν ( | ω | /c ) ~ → ~ c π X ν Z ∞ dκ ln λ ν ( κ ) → − ~ c π X ν Z ∞ κdκF ( κcω ) ddκ ln λ ν ( κ ) (7)where in the second step we took the zero-temperaturelimit according to the rule P ω → ( ~ /T ) R dω/ π anddropped contributions that are independent of geometry.In the third representation, a monotonic cutoff function F ( y ) rapidly decaying for y > F (0) = 1is made explicit; the cutoff prescription, ~ cκ/ ~ ω/ → ( ~ ω/ F ( ω/ω ), reflects penetrability of the boundary tohigh-energy modes[10]. In most of the formulas below thecutoff function is suppressed and invoked only as needed;analysis of such cases is conducted for an arbitrary cutofffunction. We note that the last representation of (7) hintsat a relationship of our approach to a contour integralmethod of calculation of the Casimir energy [7]. A. Planar geometry
Consider three Dirichlet planes at z = 0, z = a , and z = L , where z is one of the axes of the d -dimensionalrectangular coordinate system and 0 < a < L . We areinterested in the Casimir pressure exerted on the mid-dle partition at z = a . The outer boundaries are fixedin place, so that there is no need to look beyond them.Since the space is uniform relative to translations paral-lel to the boundaries, the field u ω ( r ) is expanded into aFourier series u ω ( r ) = P q u ω q ( z ) exp i qr ⊥ where r ⊥ isthe position vector perpendicular to the z axis. Then theboundary-value problem (4) for the Fourier coefficients u ω q ( z ) becomes( d dz − q − ω c ) u ω q = 0 , u ω q | ,L = 0 , u ω q | a = f ω q (8)The particular solution to (8) is u ω q ( z ) = f ω q sinh( | κ | z )sinh( | κ | a ) , z a, κ = q + ω c u ω q ( z ) = f ω q sinh( | κ | ( L − z ))sinh( | κ | ( L − a )) , a < z L (9)Substituting this in Eq.(5) we see that only the partitionat z = a contributes to the action S E with the result S E = ~ A T X ω, q | κ | (coth( | κ | a ) + coth( | κ | ( L − a ))) | f ω q | (10)where A is the macroscopic ( d − λ becomes small) for large κ ,because for high frequencies or large transverse wavevec-tor the disturbance introduced by the fluctuating bound-ary condition is localized at the boundary, to within alength that is proportional to λ itself. This will lead inwhat follows to a divergent surface energy.The Casimir energy per unit area can be deduced fromEq.(10) according to the rule Eq.(7), with the result EA = − T A X ω, q ln (cid:18) coth( | κ | a ) + coth( | κ | ( L − a ))2 πT / ~ A| κ | (cid:19) → − ~ Z dωd d − q (2 π ) d ln (coth( | κ | a ) + coth( | κ | ( L − a )))= − ~ cK d Z ∞ κ d − dκ ln (coth( κa ) + coth( κ ( L − a )))(11)where in taking the macroscopic limit we used the rule P q → A R d d − q/ (2 π ) d − . Additionally, in going fromthe first to the second representation we took the T = 0limit and dropped all a -independent contributions notinfluencing the pressure on the partition. Hereafter allsuch contributions will be systematically dropped; how-ever, we should note that in the present case part ofwhat has been omitted is a divergent integral represent-ing an infinite surface energy. The parameter K d in thethird representation is the surface area of a d -dimensionalunit sphere, S d = 2 π d/ / Γ( d/ π ) d . TheCasimir pressure on the boundary, P = − ∂ ( E / A ) /∂a canbe found in closed form P = d Γ( d +12 ) ζ ( d + 1)(4 π ) d +12 ~ c (cid:18) L − a ) d +1 − a d +1 (cid:19) (12)where Γ( x ) and ζ ( x ) are Euler’s and Riemann’s gammaand zeta functions, respectively [13]. In arriving at(12) we used the gamma function duplication formula,Γ( z )Γ( z + ) = 2 − z √ π Γ(2 z ) [13] and the value of theintegral [14] Z ∞ x d (coth x − dx = 2 − d Γ( d + 1) ζ ( d + 1) (13)We see that the partition at z = a is attractedto the closest outer boundary; specifically, in one di-mension Eq.(12) reduces to the well-known result [15].Since the outer boundaries impose the Dirichlet bound-ary conditions we can imagine joining them together.Then Eq.(12) describes Casimir interaction between twoboundaries; taking the L → ∞ limit we then reproducethe result of Ambjørn and Wolfram [5]. B. Circular geometry
Consider a Dirichlet circle of radius a in two spatialdimensions. The boundary-value problem (4) for thisgeometry becomes (cid:18) ρ ∂∂ρ ( ρ ∂∂ρ ) + 1 ρ ∂ ∂ϕ − ω c (cid:19) u ω = 0 , u ω | a = f ω ( ϕ )(14)where ρ and ϕ are the polar coordinates. Seeking the par-ticular solution in the form, u ω ( ρ, ϕ ) = R ω ( ρ ) exp inϕ ,where n is an arbitrary integer, we find that the radialfunction R ω ( ρ ) satisfies the equation d R ω dρ + 1 ρ dR ω dρ − ( ω c + n ρ ) R ω = 0 (15)whose linearly-independent solutions are modified Besselfunctions I n ( | ω | ρ/c ) and K n ( | ω | ρ/c ) [13]. Thus the par-ticular solution to the boundary-value problem (14) finiteat ρ = 0 and decaying as ρ → ∞ is u ω = ∞ X n = −∞ I n ( | ω | ρ/c ) I n ( | ω | a/c ) f ωn exp inϕ, ρ au ω = ∞ X n = −∞ K n ( | ω | ρ/c ) K n ( | ω | a/c ) f ωn exp inϕ, ρ > a (16)At the circle ρ = a both of these reduce to f ω ( ϕ ) = P ∞ n = −∞ f ωn exp inϕ . Substituting the solution (16) inEq.(5) and performing the angular integration we find S E = aπ ~ cT X ω,n | ω | I ′ n ( | ω | a/c ) I n ( | ω | a/c ) − K ′ n ( | ω | a/c ) K n ( | ω | a/c ) ! | f ωn | (17)We observe that for large n or large ω the prefactor in thissum is large, because the functions Eq.(16) are localizednear the circle boundary. This result can be simplified byapplication of the Wronskian relationship K n ( z ) I ′ n ( z ) − I n ( z ) K ′ n ( z ) = 1 /z [13]. The Casimir energy E and thepressure P = − (2 πa ) − ∂ E /∂a are then implied by Eq.(7) E = ~ c π ∞ X n = −∞ Z ∞ dκ ln ( I n ( κa ) K n ( κa )) (18) P = − ~ c π a ∞ X n = −∞ Z ∞ xdx ddx ln( I n ( x ) K n ( x )) (19)The results (18) and (19) are due to Sen [10] (see alsoRefs. [6, 16]). C. Cylindrical geometry
The Casimir interaction for three-dimensional in-finitely long Dirichlet cylinder of radius a can be in-ferred from corresponding results for the circle. First,expand the field u ω ( r ) into a Fourier series, u ω ( ρ, ϕ, z ) = P q u ωq z ( ρ, ϕ ) exp iq z z where ρ , ϕ , and z are cylindricalcoordinates, and the z axis coincides with that of thecylinder. The Fourier coefficients u ωq z ( ρ, ϕ ) are solu-tions to the boundary value problem (14) except that therole of ω /c is played by q z + ω /c . Thus cylindricalanalogs of Eqs.(15)-(17) can be written down by replac-ing the summation index ω by the combination ( ω, q z ),and ω /c by κ = q z + ω /c . Moreover, an extra fac-tor of the macroscopic cylinder length L appears in thecounterpart to Eq.(17). As a result the Casimir energyper unit length E / L and the pressure on the boundary P = − (2 πa L ) − ∂ E /∂a are given by EL = ~ c π ∞ X n = −∞ Z ∞ κdκ ln ( I n ( κa ) K n ( κa )) (20) P = − ~ c π a ∞ X n = −∞ Z ∞ x dx ddx (ln( I n ( x ) K n ( x ))) (21)respectively. Again, these results are known [17]. D. Spherical geometry in three dimensions
Consider a Dirichlet sphere of radius a in three spatialdimensions. The boundary-value problem (4) for thisgeometry becomes n r ∂∂r ( r ∂∂r ) + 1 r (cid:18) θ ∂∂θ (sin θ ∂∂θ ) + 1sin θ ∂∂ϕ (cid:19) − ω c o u ω = 0 , u ω | a = f ω ( θ, ϕ ) (22)where r , θ and ϕ are spherical coordinates. Seek-ing the particular solution in the form, u ω ( r, θ, ϕ ) = R ω ( r ) Y lm ( θ, ϕ ) where Y lm ( θ, ϕ ) is a spherical functionof order l , m [18], we find that the radial function R ω ( r )satisfies the equation d R ω dr + 2 r dR ω dr − (cid:18) ω c + l ( l + 1) r (cid:19) R ω = 0 (23)whose linearly-independent solutions are I l + ( | ω | r/c ) / √ r and K l + ( | ω | r/c ) / √ r [19]. Thusthe particular solution to the boundary-value problem(22) finite at the origin r = 0 and decaying as r → ∞ is u ω = ∞ X l =0 l X m = − l √ aI l + ( | ω | r/c ) √ rI l + ( | ω | a/c ) f ωlm Y lm ( θ, ϕ ) , r au ω = ∞ X l =0 l X m = − l √ aK l + ( | ω | r/c ) √ rK l + ( | ω | a/c ) f ωlm Y lm ( θ, ϕ ) , r > a (24)At the sphere surface r = a both of these reduce to f ω ( θ, ϕ ) = P l,m f ωlm Y lm ( θ, ϕ ). Substituting the solu-tion (24) in Eq.(5) and performing angular integrationwe find S E = ~ a T X ω,l,m | f ωlm | I l + ( | ω | a/c ) K l + ( | ω | a/c ) (25)In arriving at (25), similar to the circular case, we usedthe Wronskian of the I l +1 / ( z ) and K l +1 / ( z ) pair as wellas the property of orthogonality of the spherical functions[18]. As a result the Casimir energy E and the pressureon the boundary P = − (4 πa ) − ∂ E /∂a are implied byEq.(7) E = ~ c π ∞ X l =0 (2 l + 1) Z ∞ dκ ln I l + ( κa ) K l + ( κa ) κa (26) P = − ~ c π a ∞ X l =0 (2 l + 1) Z ∞ xdx ddx ln I l + ( x ) K l + ( x ) x (27)These equations reproduce the results due to Bender andMilton [6] who also derived more general relationships fora Dirichlet sphere in space of arbitrary dimension. E. Spherical geometry in arbitrary number ofdimensions
As a last demonstration of the efficiency of our tech-nique we derive the expressions for the Casimir energyand pressure for a Dirichlet sphere of radius a in ar-bitrary space dimension. Before proceeding we remindthe reader [20, 21, 22] that in d dimensions spheri-cal functions can be arrived at by first enumerating allhomogeneous linearly-independent harmonic polynomi-als of degree l , P l,d ( r ). After the transformation tospherical coordinates x = r cos ϕ , x = r sin ϕ cos ϕ ,..., x d − = r sin ϕ sin ϕ ... sin ϕ d − cos ϕ d − , x d = r sin ϕ sin ϕ ... sin ϕ d − sin ϕ d − , the polynomials takethe form P l,d ( r ) = r l Y l,d ( θ ), where Y l,d is a d -dimensionalspherical harmonic of degree l and θ refers to the pointon unit sphere with angular coordinates ϕ , ϕ ,..., ϕ d − .For l = 0 there exist m l,d = (2 l + d − d + l − d − l ! (28)linearly-independent spherical harmonics of order l . Fol-lowing Mikhlin [22], these will be denoted by Y ( m ) l,d ( θ ), m = 1 , , ..., m l,d . For any l the functions Y ( m ) l,d ( θ ) canbe made orthonormal on the unit sphere, and we assumethis is the case.The remaining steps mirror our discussion of the two-and three-dimensional cases. The boundary-value prob- lem (4) that has to be solved is (cid:18) ∂ ∂r + d − r ∂∂r + 1 r △ θ − ω c (cid:19) u ω = 0 , u ω | a = f ω ( θ )(29)where △ θ stands for a multiplicative angular piece ofthe Laplacian; explicit expressions for △ θ in two andthree dimensions are given in Eqs.(14) and (22), re-spectively. Seeking the particular solution in the form, u ω ( r, θ ) = R ω ( r ) Y ( m ) l,d ( θ ), we find that the radial function R ω ( r ) satisfies the equation d R ω dr + d − r dR ω dr − (cid:18) ω c + l ( l + d − r (cid:19) R ω = 0 (30)where we employed the result △ θ Y ( m ) l,d ( θ ) = − l ( l + d − Y ( m ) l,d ( θ ) [20, 21]. Linearly-independent solutions to(30) are r − d − I l + d − ( | ω | r/c ) and r − d − K l + d − ( | ω | r/c )[23] thus implying that the particular solution to theboundary-value problem (29) finite at the origin r = 0and decaying as r → ∞ is u ω = ∞ X l =0 m l,d X m =1 a d − I l + d − ( | ω | r/c ) r d − I l + d − ( | ω | a/c ) f ( m ) ωl Y ( m ) l,d ( θ ) , r au ω = ∞ X l =0 m l,d X m =1 a d − K l + d − ( | ω | r/c ) r d − K l + d − ( | ω | a/c ) f ( m ) ωl Y ( m ) l,d ( θ ) , r > a (31)At the sphere surface these reduce to f ω ( θ ) = P l,m f ( m ) ωl Y ( m ) l,d ( θ ) which is a series expansion in sphericalharmonics of an arbitrary function defined on a sphere[21, 22]. Substituting the solution (31) in Eq.(5) andperforming the integration we find S E = ~ a d − T X ω,l,m | f ( m ) ωl | I l + d − ( | ω | a/c ) K l + d − ( | ω | a/c ) (32)As a result the Casimir energy E and the pressureon the boundary P = − ( S d a d − ) − ∂ E /∂a (here S d =2 π d/ / Γ( d/
2) is the surface area of a unit sphere) aregiven by E = ~ c π ∞ X l =0 m l,d Z ∞ dκ ln I l + d − ( κa ) K l + d − ( κa )( κa ) d − (33) P = − ~ c πS d a d +1 ∞ X l =0 m l,d × Z ∞ xdx ddx ln I l + d − ( x ) K l + d − ( x ) x d − (34)which are the results due to Bender and Milton [6]; specif-ically the expression for the pressure is identical to theirEq.(3.5).In three dimensions Eqs.(33) and (34) clearly reduceto Eqs. (26) and (27). In order to see that the two-dimensional results are also reproduced, we recall that[22] for d = 2 and l > l , namelythe real and imaginary parts of ( x + ix ) l , while for d = 2 and l = 0 there is only one polynomial whichis a constant. This implies that in two dimensions thedegeneracy factor (28) satisfies the rules, m l, = 2 for l > m , = 1. With this in mind we see that for d = 2 Eqs.(33) and (34) reproduce Eqs.(18) and (19). III. DIVERGENCES AND THEIRGEOMETRICAL INTERPRETATION
Although Eq.(12) gives a finite universal Casimir pres-sure for the planar version of the problem, its circular(19), cylindrical (21), spherical (27), (34) (and any curvi-linear) counterparts are divergent. Various techniqueshave been successfully used to remove the divergences inthe case of a cylinder [17] and in spherical [6] geome-tries of odd dimensionality predicting universal Casimirpressures. However no consistent removal procedure wasfound capable of handling the spherical geometry of evenspace dimension [6]. This includes the experimentally rel-evant two-dimensional case where as an alternative, Sen[10] proposed to view the action (1) as an effective low-energy theory to be supplemented by a cutoff functionlike in Eq.(7). This removes the divergence, and leads toa finite non-universal effect.In order to understand the difference between the pla-nar and curved geometries we notice that the Casimirforce is the change of the energy upon infinitesimaldisplacement of the boundary. A geometrically sharpboundary possesses a divergent energy per unit area,coming from the exclusion of the high frequency andshort wavelength modes from an increasingly narrow re-gion near the surface. This divergent self-energy doesnot contribute to the Casimir force in the planar casebecause the overall area remains fixed as the boundary isdisplaced. However this is not the case for curved bound-aries. Indeed a change of the radius of a circle impliesa change of the perimeter and as a result the divergentself-energy will contribute into the force. This idea orig-inally due to Deutsch and Candelas[9] was recently re-expressed by Graham, Jaffe and co-workers [24] and byBarton [25]. The implication is that a real-world curvedboundary may be responsible for large non-universal por-tion of the physically measurable Casimir force [26]. Thisis consistent with Sen’s observation [10] that the coef-ficients of the non-universal terms contributing into theCasimir energy are geometrical objects such as the lengthof the circular boundary, etc. We also note that in thehigh-temperature limit the Casimir free energy can beshown to have entirely geometrical nature - it is expressedin terms of a surface integral of a quadratic function oflocal curvature [27]. a β Figure 1: Wedge of opening angle β with superimposed arcof radius a in two dimensions. The geometry-dependent contributions to the Casimireffect can be classified into three groups: (i) non-universal terms, which diverge as a power of the cutofffrequency ω ; (ii) nearly-universal terms, which divergeas a logarithm of ω ; and (iii) universal terms, whichremain finite as ω → ∞ . The nearly-universal contri-butions are characterized by universal amplitudes and aweak dependence on the choice of the cutoff function.A further test of the idea of Refs.[9, 24, 25] would con-sist in calculation of the effect in a curved geometry wherevarious contributions into the Casimir interaction can beunambiguously separated. A natural candidate is a sys-tem characterized by more than one macroscopic lengthscale where the area or circumference of the boundary isindependent of its curvature. A. Wedge-circular arc geometry in two dimensions;Dirichlet case
Consider a wedge of opening angle β with superim-posed arc of radius a , Fig. 1. This is a geometry inwhich both a Casimir force and torque exist.The semi-circular ( β = π ) version of this problem hasbeen studied earlier [28] using the zeta-function regular-ization technique. Similar to the circular case, a diver-gence was found which led the authors to conclude that”for obtaining the physical result an additional renormal-ization is needed”.The wedge-arc configuration with an arbitrary angle β has been considered in Ref. [29] with the aim of revealingthe regularities in the boundary non-smoothness contri-butions to the heat kernel coefficients. The local charac-teristics of the vacuum have been discussed by Sakharianand collaborators for a scalar field with Dirichlet bound-ary condition in general space-time dimension [30] andfor the electromagnetic field in three-dimensional space[31].Although the Casimir interaction can be obtained fromthe Casimir energy density of Ref.[30], our approach pro-vides a quicker route to the final result. Indeed theCasimir energy can be inferred from the results for thecircular geometry since the boundary-value problem weneed to solve is closely related to (14). The differenceis that we seek a solution inside the Dirichlet wedge0 ϕ β , thus implying u ω ( ρ, ϕ ) = R ω ( ρ ) sin( πnϕ/β ), n = 1 , , ... for the particular solution. The radial func-tion R ω ( ρ ) satisfies the same Eq.(15) with n being re-placed by πn/β . As a result the solution in question canbe obtained from Eq.(16) by replacing the order n of theBessel functions with πn/β , the angular function exp inϕ with sin( πnϕ/β ), and restricting the summation over n from unity to infinity. The calculation of the Casimir en-ergy is similar to that for the circular geometry with theresult E = ~ c πa ∞ X n =1 Z ∞ dx ln I πnβ ( x ) K πnβ ( x ) β → − ~ c s ∞ X n =1 Z ∞ ntdtF ( πnctω s ) ddt ln I πnβ ( πnβ t ) K πnβ ( πnβ t ) β (35)where in the second representation we restored the cutofffunction (see Eq.(7)) and introduced the length of the arc s = βa . We then employ the uniform asymptotic n ≫ n (1 + y ) / I n ( ny ) K n ( ny ) = 1 + 18 n n
11 + y − y ) + 5(1 + y ) o + O ( 1 n ) (36)which can be used to evaluate the energy (35) in the β ≪ E (1) = ~ c s ∞ X n =1 Z ∞ ntdtF ( πnctω s ) ddt ln (cid:16) πn (1 + t ) / (cid:17) = ~ c πs Z ∞ xdxF ( cxω s ) ∞ X n =1 xx + π n = ~ c πs Z ∞ dxF ( cxω s ) ( x (coth x − − x ) (37)where in going from the second to the third representa-tion we used Euler’s sum ∞ X n =1 x + ( πn ) = 12 x (cid:18) coth x − x (cid:19) (38)Although the expression (37) is divergent in the ω → ∞ limit, it is important to realize that it depends on a and β only through the arc length s = βa . In order to inter-pret (37) geometrically we note that the dimensionlesscombination ω s/c entering the argument of the cutofffunction is a ratio of the macroscopic s and microscopic c/ω length scales of the problem, and naturally one has s ≫ c/ω which is the range of applicability of the the-ory. Then in order to evaluate the first term of (37) we expand the cutoff function in a Taylor series around zero.The leading F (0) = 1 term of the expansion then givesan integral of the form (13), thus generating a universal π ~ c/ s contribution to the Casimir energy. All higherorder contributions of the expansion are convergent andproportional to negative powers of the cutoff frequency -in the ω s/c ≫ ~ ω , while the contribution from the last term is linearin the arc length. As a result we arrive at E (1) = αs + π ~ c s , α = ~ ω πc Z ∞ tF ( t ) dt (39)where α is the line tension coefficient; as expected, it is ofthe order of the microscopic energy scale ~ ω divided bythe microscopic length scale c/ω . If we specialize to thecase of an exponential cutoff function ( F ( t ) = exp( − t )),then our expression for the line tension reproduces Sen’sresult α = ~ ω / (4 πc ) [10].Even though Eq.(39) is the leading order term of the β ≪ s = βa since the radius a can be arbitrary. We see that to the leading order thereis only the dependence on the segment length (thus im-plying locality) but not on its curvature. Eq.(39) alsopredicts that if the system were given the freedom tochoose the optimal value of s by minimizing the energy,this would be a microscopic s ∼ c/ω which is a com-promise between the energy decrease due to shrinking ofthe segment length (the non-universal term) and repul-sive universal piece of the Casimir interaction preferringlarge segment lengths. The precise value of the optimal s implied by (39) is not to be trusted as s ∼ c/ω is atthe verge of applicability of the theory. If we take thesimultaneous β → a → ∞ limit but keep the length s = βa fixed, the geometry of Fig.1 turns into that ofan infinite strip of width s whose sides are connected bya straight Dirichlet bridge; its Casimir energy is givenby Eq.(39). From this viewpoint the universal π ~ c/ s piece of (39) can be understood by approximating thebridge by a strictly one-dimensional Dirichlet interval.The Casimir energy of a free field confined to an intervalof length s is − π ~ c/ s and attractive [15]. In our caseit is repulsive as these fluctuations are eliminated by theboundary. Additionally, the magnitude of our effect issmaller since the one-dimensional approximation incor-rectly assumes an infinitely sharp field localization andthus a larger change of the vacuum energy.In order to improve on Eq.(39) we subtract E (1) ,Eq.(37), from Eq.(35). The outcome U = E − E (1) isexpected to be universal or nearly-universal since thelargest non-universal fraction of the effect is already in-cluded in E (1) : U = ~ c s ∞ X n =1 n Z ∞ dt ln(2 πnβ (1+ t ) I πnβ ( πnβ t ) K πnβ ( πnβ t ))(40)where we suppressed the cutoff function. Using the ex-pansion (36) again we evaluate the energy (40) to thenext order in β ≪
1. If the integration is performed first,then we find U = − ~ cβ πs ∞ X n =1 n − , β ≪ , (41)which is marginally divergent. This means that settingthe cutoff function at unity is not justified, and that thereis a weak dependence on the cutoff frequency ω . There-fore we restore the cutoff function, go beyond the leadingorder term in the Debye expansion (36) and write Eq.(40)as U = − ~ cβ πs Z ∞ xdxF ( cxω s ) ddx ∞ X n =1 n x + ( πn ) − πn ) ( x + ( πn ) ) + 5( πn ) ( x + ( πn ) ) o (42)The sums over n can be computed with the help of Euler’sformula (38) and two consequent relationships: ∞ X n =1 ( πn ) ( x + ( πn ) ) = ( x coth x ) ′ x (43) ∞ X n =1 ( πn ) ( x + ( πn ) ) = ( x ( x coth x ) ′ ) ′ x (44)where the prime stands for the derivative with respect to x . Comparing the right- and left-hand sides of Eqs.(38),(43), and (44) we observe that the sum P n ... o in Eq.(42)approaches a constant limit as x →
0. This impliesthat the integral over x is convergent at the lower limit.On the other hand, in the x → ∞ limit we find that P n ... o → − / (2 x ) with the implication that if the cut-off function is set to unity, then, consistent with Eq.(41),the integral over x would logarithmically diverge at theupper limit. The role of the cutoff function consists inrestoring the convergence by effectively setting the up-per limit of the integration at ω s/c ≫
1. In view ofthe logarithmic character of the divergence, the outcomeis rather insensitive to the upper integration limit. Asa result we arrive at the nearly-universal contribution tothe Casimir energy U = − ~ cβ πs ln ω sc , β ≪ P Nn =1 n − = C + ln N + ǫ N where C isEuler’s constant and ǫ N → N → ∞ [13]. The param-eter N is estimated by recalling that the suppressed cutofffunction F ( πnct/ω s ) effectively ends the sum of 1 /n at n = N such as πnct/ω s ≈ t , as implied by the second-order term of the Debye ex-pansion (36), is of the order unity. Thus N ≈ ω s/c ≫ fixed arc length s whenthe leading order contribution (39) can be regarded asa constant, the zero-point motion induces a widening torque − ∂ U /∂β which could be expected on physicalgrounds. Combining Eqs.(39) and (45) we arrive at theexpression for the Casimir energy of the Dirichlet wedgeof arc length s and opening angle β = s/a ≪ E = αs − ~ cs πa ln ω sc + π ~ c s (46)This result supports a geometrical interpretation forthe non-universal contributions to the Casimir energy.Indeed, when the frequency cutoff ω tends to infin-ity, the strongest ( ω ) divergence comes from the firstterm of (46) proportional to the length of the bound-ary thus supporting the idea of Refs.[9, 24, 25]. Thesub-leading logarithmically divergent term of (46) canbe also understood geometrically as being proportionalboth to the length of the boundary and to the squareof its curvature. This term can be alternatively inter-preted as a finite-size correction to the surface tension: α → α − ~ c ln( ω s/c ) / πa . We note that Eq.(46) hasa local character and exhibits a term-by-term correspon-dence with its circular counterpart [10].The accuracy of the Debye expansion (36), implies thatEq.(46) approximately captures the whole 0 β π range. B. Wedge-circular arc geometry in two dimensions:periodic boundary conditions
The Casimir effect is sensitive to changes in spacetopology. In our next example we demonstrate that thenon-universal contributions to the Casimir effect do notseem to be sensitive to topology while the universal onesare. Consider the wedge-circular arc geometry as in Fig.1 but assume that the wedge edges are physically iden-tical. This is implemented by imposing the condition ofperiodicity, u ( ρ, ϕ ) = u ( ρ, ϕ + β ), where the opening an-gle β can now be arbitrary. This means that the field u belongs to a conical surface with a Dirichlet circle whichis a distance a away from cone’s apex as measured alongthe cone surface [29]. For β = 2 π the geometry becomesthat of a plane with the Dirichlet circle.The Casimir energy of this configuration can be in-ferred from the solutions of the circular and wedge cases.Indeed, the boundary-value problem we need to solveis posed by Eq.(14) with the boundary field satisfyingthe condition of periodicity, f ω ( ϕ ) = f ω ( ϕ + β ). Thisimplies u ω ( ρ, ϕ ) = R ω ( ρ ) sin(2 πnϕ/β ), n = 1 , , ... or u ω ( ρ, ϕ ) = R ω ( ρ ) cos(2 πnϕ/β ), n = 0 , , , ... for theparticular solution. In both cases the radial function R ω ( ρ ) satisfies the same Eq.(15) with n being replacedby 2 πn/β . The Casimir energy can then be inferred fromEq.(35) as E = ~ c πa Z ∞ dx ln I ( x ) K ( x ) β + ~ cπa ∞ X n =1 Z ∞ dx ln I πnβ ( x ) K πnβ ( x ) β (47)where the first term is due to the fluctuating angle-independent Fourier component of the boundary field f ω while the sum is a contribution from the angle-dependent components. The latter are represented by asine and a cosine Fourier series contributing equally intothe Casimir energy. We note that for β = 2 π Eq.(47)reduces to its circular counterpart, Eq.(18).In order to understand the geometry dependence of thefirst term of (47) we restore the cutoff function F ( y ): E = − ~ c πa Z ∞ xdxF ( cxω a ) ddx ln I ( x ) K ( x ) β (48)The derivative removes the dependence on the openingangle β . Since 2 xI ( x ) K ( x ) ≈ ω a/c thus leading to a geometry-independent estimate E ∼ ~ ω . The a -dependent partcan be found by subtracting from (48) the same integralwith I ( x ) K ( x ) replaced with its large argument limit1 / x : E → ~ c πa Z ∞ dx ln 2 xI ( x ) K ( x ) = − . ~ c πa (49)This is the universal piece of the Casimir effect when thecutoff function is set at unity (i. e. the ω a/c → ∞ limitis taken). The integral was numerically evaluated by Sen[10] and more accurately by Milton and Ng [16] whosevalue is used in (49).The second term of Eq. (47) can be analyzed in amanner similar to that of Eq.(35). The results of suchanalysis combined with (49) give the Casmir energy forthe wedge-arc geometry with periodic boundary condi-tions: E = αs − ~ cs πa ln ω sc + π ~ c s − . ~ c πa (50)where the coefficient of surface tension is defined inEq.(39). We see that the cutoff-dependent terms ofEqs.(46) and (50) are the same. At the same time theeffect of topological change in going from the Dirich-let to periodic boundary conditions manifests itself inthe universal parts of the Casimir effect. Indeed, the s -dependent piece of the interaction, π ~ c/ s , is fourtimes larger than its Dirichlet counterpart, π ~ c/ s , inEqs.(39) and (46). This mirrors the relationship betweenthe Dirichlet and periodic Casimir energies for the free field in one dimension [33]. Additionally, the last attrac-tive term of (50) is unique to the periodic geometry.For the special case of a circle in a plane, s = 2 πa , Eq.(50) simplifies to E = 2 παa − ~ c a ln ω ac + 0 . ~ c a (51)This agrees with Sen’s result [10] except for the magni-tude of the last term. We believe Sen’s value 0.045 is atypographical error (according to the numerical resultshe quotes, the value should be 0.054).Similar to the argument following Eq.(39) we can takethe simultaneous β → a → ∞ limit but keep thesegment length s = βa fixed. Then Eq.(50) simplifies to E (1) = αs + π ~ c s (52)which is a periodic counterpart of Eq.(39); it gives theCasimir energy of a Gaussian field defined on an infinitecylinder of radius s/ π in the presence of a Dirichlet circlebelonging to the cylinder surface. IV. ARBITRARY SMOOTH BOUNDARY OF ACOMPACT REGION
Since for the circular geometry the length of the bound-ary and its curvature are not independent, a route differ-ent from direct calculation is needed to demonstrate thegeometrical nature of the non-universal contributions tothe Casimir interaction, Eq.(51). For that purpose Sen[10] employed a result due to Pleijel [34] which relatesasymptotic behavior of a certain integral of a Green’sfunction to geometry. In the process Sen derived a re-markable expression for the non-universal part of theCasimir energy E ( ω ) for an arbitrary smooth boundaryof a compact region. Here we show how this result canbe understood heuristically by employing our findings forthe wedge geometry.First we notice that for the wedge whose edges areDirichlet/periodically connected by an infinitesimallysmall arc/circle of length ds the non-unversal part of theCasimir energy (see Eqs. (41), (46) and (50)) can berewritten as a differential relationship d E ( ω ) = αds + d U = αds − ~ c π C ( s ) ds ∞ X n =1 n − (53)where C = 1 /a is the curvature of the arc. Our calcula-tions indicate that the energy (53) has to be attributedto the arc itself rather than to the conditions imposedon the wedge edges; those conditions determine the uni-versal part of the Casimir effect. With this in mind letus consider an arbitrary smooth boundary of a compactregion. Then Eq.(53) can be interpreted as representinga non-universal contribution into the Casimir energy dueto the infinitesimally small boundary element ds whose0position, the length s along the boundary, is measuredrelative to an arbitrary reference point and C ( s ) is thecurvature at location s .For an arbitrary smooth boundary the non-universalpiece of the Casimir energy can be found by integrating(53) over the length of the boundary E ( ω ) = αS − (cid:18) ~ c π Z C ( s ) ds (cid:19) ∞ X n =1 n − (54)where S = R ds is the total length of the boundary. Wenotice that the marginally divergent sum of n − does nothave any reference to specific geometry. This leads us to a conjecture that (similar to the case of the circular arc) theeffect of the frequency cutoff ω consists in replacementof the divergent sum in (54) with the finite logarithm ofthe large ratio of the macroscopic and microscopic lengthscales: E ( ω ) = αS − (cid:18) ~ c π Z C ( s ) ds (cid:19) ln ω Sc (55)Since Eq.(55) has logarithmic accuracy, the precise mean-ing of the macroscopic length scale is not essential - wehave chosen it to be the total length S of the boundary.The result (55) is due to Sen [10, 35]. Our heuristic argu-ment which leads to Sen’s result (55) also links the Pleijelformula [34] to the Debye expansion (36) which is in theheart of our wedge results.The remarkable feature of Eq.(55) is that it is solelydetermined by local geometry of the boundary. In ar-riving at (55) we employed additivity of the interactionwhich is known generally not to be the case for Casimirinteractions. We note however that the additivity onlyholds for the strictly non-universal piece of the effect. Forthe nearly-universal second term of Eq.(55) the additiv-ity is logarithmically weakly violated. This allowed usto proceed by assuming the additivity and to deal withthe logarithmic divergence separately. Additionally, our calculations indicate that additivity does not hold for theuniversal parts of the Casimir energy which could not beexpressed in a differential form. The implication is thatit would be very difficult if not impossible to come upwith a purely geometrical formula for the universal partof the Casimir effect. V. SUMMARY
To summarize, we have demonstrated how Casimir ef-fects caused by sharp boundaries can be efficiently com-puted by focusing on the quantum fluctuations elimi-nated by these boundaries. The applicability of thismethod is not limited to the scalar field theory (1),Dirichlet boundaries, zero-temperature limit or to the ge-ometries we have considered.Second, we presented an explicit calculation to supportthe idea [9, 24, 25] that the divergent Casimir forces en-countered in the presence of curved boundaries have ge-ometrical origin - they are largely due to divergent self-energy contributions. Our analysis also supports Sen’sproposal [10] that the Casimir effect in a Dirichlet ring intwo dimensions is finite and non-universal with the cutofffrequency ω supplied by the properties of the materialthe boundary is made of. More work is needed to fur-ther explore the geometrical nature of the non-universalparts of the Casimir effect as they may be responsiblefor largest contributions into experimentally measurableCasimir force. VI. ACKNOWLEDGMENTS
We thank A. A. Saharian for informing us of his workon a related topic. This work was supported by theThomas F. Jeffress and Kate Miller Jeffress MemorialTrust. [1] H. B. G. Casimir,
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Statistical Physics ,vol.V, Part 1, third edition, revised and enlarged by E.M. Lifshitz and L. P. Pitaevskii (Pergamon, 1980), p.170. [15] T. H. Boyer, Am. J. Phys. , 990 (2003), and referencestherein.[16] K. A. Milton and Y. J. Ng, Phys. Rev. D , 842 (1992).[17] V. V. Nesterenko and I. G. Pirozhenko, J. Math. Phys. , 4521 (2000).[18] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1977), Section 28.[19] An equation which is only different from (23) in the signof the ω /c combination is solved in Section 33 of Ref.[18].[20] A. Sommerfeld, Partial Differential Equations in Physics (Academic Press Inc. Publishers, New York, N. Y. 1949),Chapter V, Appendix IV.[21] C. M¨uller,
Spherical Harmonics , Lecture Notes in Math-ematics , (Springer-Verlag, Berlin 1966).[22] S. G. Mikhlin, Mathematical Physics, An AdvancedCourse , (North Holland Publishing Company, 1970),Chapter 12.[23] An equation which is only different from (30) in the signof the ω /c combination is solved in Section B of Ref.[20].[24] N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, M.Scandurra and H. Weigel, Nucl. Phys. B , 49 (2002);Phys. Lett. B , 196 (2003); N. Graham, R. L. Jaffeand H. Weigel, Int. J. Mod. Phys. A , 846 (2002);N. Graham and K. D. Olum, Phys. Rev. D , 085014(2003); Erratum-ibid. D , 10990 (2004); K. D. Olumand N. Graham, Phys. Lett. B , 175 (2003); R. L.Jaffe, arXiv:hep-th/0307014v2; N. Graham, R. L. Jaffe, V. Khemani, M. Quandt, O. Schroeder and H.Weigel,Nucl. Phys. B , 379 (2004).[25] G. Barton, J. Phys. A: Math. Gen. , 1011 (2004).[26] The conclusions of Graham, Jaffe and co-workers [24]were criticized in K. A. Milton, Phys. Rev. D , 065020(2003) and J. Phys. A: Math. Gen. , 6391(2004); fur-ther discussion of surface divergences can be found inMilton’s review [3] and in S. A. Fulling, J. Phys. A: Math.Gen. , 6857 (2003).[27] R. Balian and B. Duplantier, Ann. Phys. (N. Y.) ,165 (1978).[28] V. V. Nesterenko, G. Lambiase, G. Scarpetta, J. Math.Phys. , 1974 (2001), Section V.[29] V. V. Nesterenko, I. G. Pirozhenko, and J. Dittrich,Class. Quantum Grav. , 431 (2003).[30] A.A. Saharian, A.S. Tarloyan, J. Phys. A: Math. Gen. , 8763 (2005).[31] A. A. Saharian, Eur. Phys. J. C , 721 (2007).[32] K. A. Milton, L. L. DeRaad, Jr. and J. Schwinger, Ann.Phys. (N.Y.) , 388 (1978).[33] H. W. Bl¨ote, J. L. Cardy , and M. P. Nightingale, Phys.Rev. Lett. , 742 (1986); I. Affleck, Phys. Rev. Lett. ,746 (1986).[34] A. Pleijel, Arkiv. Matematik , 553 (1954); see also K.Stewartson and R. T. Waechter, Proc. Cambridge Philos.Soc. , 353 (1971) and R. T. Waechter, Proc. CambridgePhilos. Soc. , 439 (1972).[35] Sen’s result [10] is inadvertantly missing a factor of ππ